Built-in Log-Likelihood Functions

This section displays the basic formulas used by the NLMIXED procedure to compute the conditional log-likelihood functions
of the data given the random effects. Note, however, that in addition to these basic equations, the NLMIXED procedure employs
a number of checks for missing values and floating-point arithmetic. You can see the entire program used by the NLMIXED procedure
to compute the conditional log-likelihood functions by adding the LIST
debugging option to the PROC NLMIXED
statement.

This parameterization of the gamma distribution differs from the parameterization used in the GLIMMIX and GENMOD procedures.
The following statements show the equivalent reparameterization in the NLMIXED procedure that fits a generalized linear model
for gamma-distributed data in the parameterization of the GLIMMIX procedure:

This form of the negative binomial distribution is one of the many parameterizations in which the mass function or log-likelihood
function appears. Another common parameterization uses

with , .

Note that the parameter n can be real-numbered; it does not have to be integer-valued. The parameterization of the negative binomial distribution in
the NLMIXED procedure differs from that in the GLIMMIX and GENMOD procedures. The following statements show the equivalent
formulations for maximum likelihood estimation in the GLIMMIX and NLMIXED procedures in a negative binomial regression model: